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\begin{document}

\title{高等代数二}
\subtitle{7-1-线性映射 }
%\institute{上海立信会计金融学院}
%\author{王立庆}
\author{{\ppr LQW}}
\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
%\date{{\ppr 2023年3月9日} }

\maketitle

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%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{7.1.i. 作业：星期天晚上十点半之前在网络教学平台提交 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{enumerate}
\item   整理课堂笔记，补充没写完的计算或证明。
\item   习题(7.1)\#1,3,4,5，抄写题目。
\end{enumerate}

\end{frame}

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\begin{frame}{7.1.ii. 目录 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{enumerate}
\item 线性映射的定义
\item 线性映射的例子
\item 线性映射的基本性质
\item 定义：子空间在线性映射下的像与原像
\item 定理7.1.1. 子空间在线性映射之下的像与原像仍是子空间
\item 定义：线性映射的像空间与核空间
\item 定理7.1.2. 线性映射是满射和单射的充分必要条件
\item 两个线性映射的合成
\item 线性映射的逆映射
\end{enumerate}


\end{frame}

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%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{7.1.iii. 课堂讲解重点 }

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%每页详细内容

\begin{enumerate}

\item  线性映射的概念和例子
\item  子空间在线性映射之下的像与原像
\item  线性映射的像空间与核空间
\item  线性映射的合成
\item  线性映射的逆映射

\end{enumerate}

\end{frame}

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\begin{frame}{7.1.1. 线性映射的定义}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item {\color{red}问题：数域 $F$ 上的两个向量空间之间的一个映射 $\sigma:V\to W$ 称为是一个线性映射，如果它保持向量空间的线性运算。什么是保持线性运算？}

\vspace{0.3cm}

\item 解答：保持线性运算是指下述两个条件：
 \begin{enumerate}
\item 对任意 $\alpha,\beta\in V$, 都有 $\sigma(\alpha+\beta)=\sigma(\alpha)+\sigma(\beta)$. 
\item 对任意 $\alpha\in V$ 和任意 $k\in F$, 都有 $\sigma(k\alpha)=k\sigma(\alpha)$. 
\end{enumerate}

\item 注：
 \begin{enumerate}
\item 线性映射连接的两个向量空间必须是定义在同一个数域上的。
\item %上面的两个等式中，
等式左边的加法和数乘是 $V$ 中的，右边的加法和数乘是 $W$ 中的。
\end{enumerate}

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.1.2. 线性映射的例子1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：写出一个线性映射的例子，$\sigma: \mathbb{R}^2\to \mathbb{R}^3$. } 

\vspace{0.3cm}

\item 解答：这由三个二元线性函数确定。

\begin{eqnarray*}
\mathbb{R}^2 \ni
\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}
\overset{\sigma}{\mapsto}
\begin{pmatrix} x_1 \\ x_1-x_2 \\ x_1+x_2 \end{pmatrix}
\in \mathbb{R}^3.
\end{eqnarray*}
写成矩阵形式，
\begin{eqnarray*}
\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}
\overset{\sigma}{\mapsto}
\begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}=
\begin{pmatrix} 1&0 \\ 1&-1 \\ 1&1 \end{pmatrix}
\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}.
\end{eqnarray*}

\end{itemize}

\end{frame}

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\begin{frame}{7.1.3. 例子1的图像}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：画出上一页的线性映射 $\sigma: \mathbb{R}^2\to \mathbb{R}^3$ 的图像。} 

\vspace{0.3cm}

\item 解答：
\begin{enumerate}
\item  先确定向量 $(1,0)$ 和 $(0,1)$ 分别映成了什么。
\item  然后向量 $(1,0)$ 和 $(0,1)$ 的线性组合映成了什么。
\item  这个线性映射将平面映成了下述矩阵的列空间：
\begin{eqnarray*}
A=\begin{pmatrix} 1&0 \\ 1&-1 \\ 1&1 \end{pmatrix}. 
\end{eqnarray*}

\end{enumerate}

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.1.4. 例子1的拓展 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red}问题：设有向量值多元函数，
\begin{eqnarray*}
\left\{\begin{array}{rcl}
y_1 &=& f(x_1,x_2), \\ 
y_2 &=& g(x_1,x_2), \\ 
y_3 &=& h(x_1,x_2).
\end{array}\right.
\end{eqnarray*}
求它在 $(x_1,x_2)=(0,0)$ 附近的线性近似。
}

\vspace{0.3cm}

\item 解答：这是向量值多元函数在一点的泰勒展开，保留常数项和一阶导数所在的线性项。


\end{itemize}

\end{frame}


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\begin{frame}{7.1.5. 线性映射的例子2}

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%每页详细内容

\begin{itemize}
\item  {\color{red}问题：定义立体空间到$Oxy$平面的正射影， 并验证这是一个线性映射。}
%$\sigma: \mathbb{R}^3\to \mathbb{R}^2$,

\vspace{0.3cm}

\item 解答：按照取值集合的不同，这个正射影有两种理解：

\begin{eqnarray*}
\mathbb{R}^3 \ni
\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}
\overset{\sigma}{\mapsto}
\begin{pmatrix} x_1 \\ x_2 \\ 0 \end{pmatrix}
\in \mathbb{R}^3, 
\,\,\,\text{ 对应于矩阵 }\,\,\,
A=\begin{pmatrix} 1&0&0  \\ 0&1&0 \\ 0&0&0 \end{pmatrix}.
\end{eqnarray*}

\begin{eqnarray*}
\mathbb{R}^3 \ni
\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}
\overset{\sigma}{\mapsto}
\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}
\in \mathbb{R}^2,
\,\,\,\text{ 对应于矩阵 }\,\,\,
A=\begin{pmatrix} 1&0&0  \\ 0&1&0 \end{pmatrix}.
\end{eqnarray*}

\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.1.6. 线性映射的例子3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：写出从 ${\mathbb{R}}^{\,n}$ 到 $\mathbb{R}^{\,m}$ 的一个线性映射的例子。} 

\vspace{0.3cm}

\item 解答：这由一个 $m\times n$ 矩阵所确定。

\begin{eqnarray*}
{\mathbb{R}}^{\,n} \ni
\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}
\overset{\sigma}{\mapsto}
\begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_m \end{pmatrix}=
\begin{pmatrix} a_{11}&a_{12}& \cdots & a_{1n} \\ a_{21}&a_{22}& \cdots & a_{2n} \\  \vdots & \vdots & &\vdots \\ a_{m1}&a_{m2}& \cdots & a_{mn} \\  \end{pmatrix}
\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}
\in \mathbb{R}^{\,m}. 
\end{eqnarray*}

\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.1.7. 线性映射的例子4}

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%每页详细内容

\begin{itemize}
\item  {\color{red}问题：什么是零映射？}

\vspace{0.3cm}

\item 解答：将定义域中的所有向量都映成值域中的零向量的映射，即
\begin{eqnarray*}
V \ni \alpha \overset{\sigma}{\mapsto} \theta_W \in W,
\end{eqnarray*}
其中 $\theta_W$ 是向量空间 $W$ 的零向量。


%\item 

\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.1.8. 线性映射的例子5}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：什么是位似映射？}

\vspace{0.3cm}

\item 解答：位似映射是一个向量空间到自身的映射，将每个向量都映成某个固定的倍数。即存在一个数量 $k\in F$, 使得
\begin{eqnarray*}
V \ni \alpha \overset{\sigma}{\mapsto} k\alpha \in V. 
\end{eqnarray*}

%\item 

\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.1.9. 线性映射的例子6}

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%每页详细内容

\begin{itemize}
\item  {\color{red} 问题：什么是数域 $F$ 上的一个 $n$ 元线性函数，或者 $F^n$ 上的一个线性型？}

\vspace{0.3cm}

\item 解答：一个线性型是一个从 $F^{\, n}$ 到 $F$ 的线性映射，即
\begin{eqnarray*}
{F}^{\,n} \ni
\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}
\overset{\sigma}{\mapsto}
y=\begin{pmatrix} a_1 & a_2 & \cdots & a_n \end{pmatrix}
\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}
\in F.
\end{eqnarray*}
换句话说，一个线性型是这样一个表达式
\begin{eqnarray*}
y=a_1x_1 + a_2x_2 + \cdots + a_nx_n. 
\end{eqnarray*}
%\item 

\end{itemize}

\end{frame}


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\begin{frame}{7.1.10. 线性映射的例子7}

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%每页详细内容

\begin{itemize}
\item  {\color{red} 问题：证明多项式空间上的求导运算是一个线性映射。} 

\vspace{0.3cm}

\item 解答：求导运算是这样一个映射
\begin{eqnarray*}
F[x] \ni f(x) \overset{\sigma}{\mapsto} f{\,\,'} (x) \in F[x].
\end{eqnarray*}

验证这是一个线性映射。对任意的 $f_1(x),f_2(x),f(x) \in F[x]$, 对任意的 $k\in F$, 要验证下述两个等式成立，
\begin{eqnarray*}
\sigma(f_1(x)+f_2(x)) &=&  \sigma(f_1(x))+ \sigma(f_2(x)), \\
\sigma(kf(x)) &=&  k\sigma(f(x)).
\end{eqnarray*}
按照 $\sigma$ 的定义，两边分别计算，验证即得。

\item 注意：一个多项式看做一个抽象的向量。

\end{itemize}

\end{frame}


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\begin{frame}{7.1.11. 线性映射的例子8}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red} 问题：证明函数空间 $C[a,b]$ 到自身的积分变换是一个线性变换。}

\vspace{0.3cm}

\item 解答：积分运算可以看作是这样一个映射
\begin{eqnarray*}
C[a,b] \ni f(x) \overset{\sigma}{\mapsto} \sigma(f)(x) = \int_a^x f(t)dt \in C[a,b].
\end{eqnarray*}
对任意的 $f_1(x),f_2(x),f(x) \in C[a,b]$, 对任意的 $k\in\mathbb{R}$, 要验证
\begin{eqnarray*}
\sigma(f_1+f_2) &=&  \sigma(f_1)+ \sigma(f_2), \\
\sigma(kf) &=&  k\sigma(f).
\end{eqnarray*}

\item 注意：上述等式两边都是 $x$ 的函数，即 $V=C[a,b]$ 中的元素。例如，
\begin{eqnarray*}
[ \sigma(f_1+f_2) ] (x) =  \int _a^x [f_1(t)+f_2(t)] dt.
%[\sigma(f_1)+ \sigma(f_2)](x) &=& \sigma(f_1)(x) + \sigma(f_2)(x) = \int_a^x f_1(t)dt + \int_a^x f_2(t)dt. 
\end{eqnarray*}

\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.1.12. 线性映射的基本性质}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red} 问题：证明线性映射 $\sigma:V\to W$ 一定将零向量映到零向量。} 

\vspace{0.3cm}

\item 证明：任取 $\alpha\in V$, 记 $\theta_V$ 和 $\theta_W$ 分别是 $V$ 和 $W$ 的零向量，
\begin{eqnarray*}
\sigma(\theta_V) &=& \sigma(\alpha-\alpha) \\
&=& \sigma[\alpha+(-1)\alpha] \\
&=& \sigma(\alpha)+(-1)\sigma(\alpha)\\
&=& \sigma(\alpha) - \sigma(\alpha)\\
&=& \theta_W. 
\end{eqnarray*}

\item 这个证明使用线性映射必须保持线性运算的性质，和向量空间的性质：
\begin{eqnarray*}
(-1)\xi = -\xi.
\end{eqnarray*}

\end{itemize}

\end{frame}

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\begin{frame}{7.1.13. 概念：线性映射的像与原像}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red} 问题：}
\begin{enumerate}
\item  {\color{red} 什么是子空间 $V_1\subseteq V$ 在线性映射 $\sigma:V\to W$ 下的像？}
\item  {\color{red} 什么是子空间 $W_1\subseteq W$ 在线性映射 $\sigma:V\to W$ 下的原像？} 
\end{enumerate}

\vspace{0.3cm}

\item 解答：子空间的像与原像的定义分别是
\begin{eqnarray*}
\sigma(V_1) &=& \{\sigma(\alpha) \mid \alpha\in V_1\}, \\
\sigma^{-1}(W_1) &=& \{\alpha\in V \mid \sigma(\alpha)\in W_1\}. 
\end{eqnarray*}

\end{itemize}

\end{frame}

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\begin{frame}{7.1.14. 映射的像与原像}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red} 问题：设有集合之间的映射
%这两个定义跟集合之间的映射，子集的像与原像是一样的。例如，
\begin{eqnarray*}
\mathbb{R}\ni x\mapsto f(x)=x^2\in \mathbb{R}.
\end{eqnarray*}
求子集（闭区间） $[-1,2]$ 的像和原像。}

\vspace{0.3cm}

\item  解答：像和原像分别为
\begin{eqnarray*}
f\, ([-1,2]) = [0,4], \,\,\,
f^{\,\,-1}([-1,2]) = [0,\sqrt{2}].
\end{eqnarray*}


\end{itemize}

\end{frame}

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\begin{frame}{7.1.15. 定理7.1.1.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red} 定理：子空间在线性映射下的像和原像都仍然是子空间。} 

\vspace{0.3cm}

\item 证明：现在证明子空间的像集也是一个子空间，故称像空间。

\begin{enumerate}
\item  设有线性映射 $\sigma:V\to W$, 设有子空间 $V_1\subseteq V$. 要证明 $f\,(V_1)$ 是 $W$ 的子空间。

\item  设有任意向量 $\beta_1,\beta_2 \in f\,(V_1)$, 要证明 $\beta_1+\beta_2\in f\,(V_1)$. 
\begin{eqnarray*}
\beta_1+\beta_2 = f(\alpha_1) + f(\alpha_2) = f(\alpha_1+\alpha_2).
\end{eqnarray*}

\item  设有任意向量 $\beta \in f\,(V_1)$ 和任意数 $k\in F$, 要证明 $k\beta\in f\,(V_1)$. 
\begin{eqnarray*}
k\beta = kf(\alpha) = f(k\alpha).
\end{eqnarray*}

\end{enumerate}

类似可证子空间的原像也是一个子空间。

\end{itemize}

\end{frame}

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\begin{frame}{7.1.16. 概念：线性映射的像与核}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red} 问题：什么是线性映射 $\sigma:V\to W$ 的像空间与核空间？} 

\vspace{0.3cm}

\item 解答：这两个空间的定义分别为
\begin{eqnarray*}
\text{Im}(\sigma) &=& \sigma(V) \,\,=\,\, \{\sigma(\alpha) \mid \alpha\in V\}, \\
\text{Ker}(\sigma) &=& \sigma^{-1}(\theta_W) \,\,=\,\, \{\alpha\in V \mid \sigma(\alpha)=\theta_W\}. 
\end{eqnarray*}
也就是说，全空间的像集称为线性映射的像空间，零子空间的原像称为线性映射的核空间。
%\item 

\end{itemize}

\end{frame}

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\begin{frame}{7.1.17. 定理7.1.2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red} 定理：设 $V$ 和 $W$ 都是数域 $F$ 上的向量空间。设 $\sigma:V\to W$ 是一个线性映射。则有：}
\begin{enumerate}
\item  {\color{red} 这个线性映射是满射当且仅当它的像空间是全空间。}
\item  {\color{red} 这个线性映射是单射当且仅当它的核空间是零子空间。} 
\end{enumerate}

\vspace{0.3cm}

\item 证明：我们来证明第2条。

\begin{enumerate}

\item[(2a)]  设 $\sigma$ 是单射。设 $\alpha\in\text{Ker}(\sigma)$. 则 $\sigma(\alpha)=\theta_W$. 但是总有 $\sigma(\theta_V)=\theta_W$, 所以根据单射的定义，只能有 $\alpha=\theta_V$. 因此 $\text{Ker}(\sigma)=\{\theta_V\}$. 

\item[(2b)]  另一方面，设 $\text{Ker}(\sigma)=\{\theta_V\}$. 现设有 $\alpha_1,\alpha_2\in V$ 使得 $\sigma(\alpha_1)=\sigma(\alpha_2)$. 于是根据 $\sigma$ 保持线性，可得 $\sigma(\alpha_1-\alpha_2)=\theta_W$, 所以 $\alpha_1-\alpha_2\in \text{Ker}(\sigma)$. 由条件 $\text{Ker}(\sigma)=\{\theta_V\}$, 所以 $\alpha_1-\alpha_2=\theta_V$,  所以 $\alpha_1=\alpha_2$. 这就证了 $\sigma$ 是单射。

\end{enumerate}


%\item 

\end{itemize}

\end{frame}

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\begin{itemize}
\item  {\color{red}问题：两个线性映射的合成仍然是一个线性映射。}

\vspace{0.3cm}

\item 证明：两个线性映射的合成形如
\begin{eqnarray*}
\begin{array}{ccccc}
V_1 &\overset{\sigma}{\to} & V_2 & \overset{\tau}{\to} & V_3  \\
\alpha &\mapsto& \sigma(\alpha) &\mapsto & \tau(\sigma(\alpha)).
\end{array}
\end{eqnarray*}
因此我们需要证明的是这样两个等式
\begin{eqnarray*}
\tau(\sigma(\alpha_1 + \alpha_2)) &=& \tau(\sigma(\alpha_1) + \tau(\sigma(\alpha_2)), \\
\tau(\sigma(k\alpha)) &=& k\tau(\sigma(\alpha)).
\end{eqnarray*}
由条件 $\sigma$ 和 $\tau$ 都保持线性运算可知。

%\item 

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.1.19. 线性映射的逆映射}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：线性映射如果存在逆映射，那么这个逆映射也是一个线性映射。} 

\vspace{0.3cm}

\item 证明：
\begin{enumerate}
\item  定义：如果一个映射 $\varphi:S\to T$ 是双射，那么称它的逆映射存在，并且定义为 $\varphi^{-1}:T\to S$, 其中 $\varphi^{-1}(t)=s$ 当且仅当 $\varphi(s)=t$.  %写成数学符号，
%\begin{eqnarray*}
%S & \overset{\varphi}{\underset{\varphi^{-1}}{\rightleftarrows}} & T \\
%s & \rightleftarrows & t
%\end{eqnarray*}

\item  现在设 $\sigma:V\to W$ 是一个线性映射，而且是一个双射。因此这时存在逆映射 $\sigma^{-1}:W\to V$. 要证明的是对任意 $\beta_1,\beta_2\in W$ 与任意 $k_1,k_2\in F$, 有
\begin{eqnarray*}
\sigma^{-1}(k_1\beta_1+k_2\beta_2) = k_1\sigma^{-1}(\beta_1)+k_2\sigma^{-1}(\beta_2). 
\end{eqnarray*}

\item  这等价于下述等式，根据 $\sigma$ 是线性映射，可知这是对的。
\begin{eqnarray*}
k_1\beta_1+k_2\beta_2 = \sigma[ k_1\sigma^{-1}(\beta_1)+k_2\sigma^{-1}(\beta_2)]. 
\end{eqnarray*}


\end{enumerate}

\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{习题(7.1)\#1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red}问题：设 $\xi=(x_1,x_2,x_3)$ 是 $\mathbb{R}^3$ 的任意向量，下列映射哪些是 $\mathbb{R}^3$ 到自身的线性映射？}
\begin{enumerate}
\item {\color{red}$\sigma(\xi)=\xi+\alpha$, 其中 $\alpha$ 是 $\mathbb{R}^3$ 的一个固定向量。} 
\item {\color{red}$\sigma(\xi)=(2x_1-x_2+x_3, x_2+x_3, -x_3)$. } 
\item {\color{red}$\sigma(\xi)=(x_1^2, x_2^2, x_3^2)$. } 
\item {\color{red}$\sigma(\xi)=(\cos(x_1), \sin(x_1), 0)$. } 
\end{enumerate}

\vspace{0.3cm}

\item 思路：按线性映射的定义验证。

\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{习题(7.1)\#3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red}问题：记 $M_n(F)$ 表示数域 $F$ 上的一切 $n$ 阶矩阵组成的向量空间。取定 $A\in M_n(F)$. 对任意 $X\in M_n(F)$, 定义 $\sigma(X)=AX-XA$. }
\begin{enumerate}
\item  {\color{red}证明 $\sigma$ 是 $M_n(F)$ 到自身的线性映射。}
\item  {\color{red}证明对于任意的 $X,Y\in M_n(F)$, 有 $\sigma(XY)=\sigma(X)Y+X\sigma(Y)$. }
\end{enumerate}

\vspace{0.3cm}

\item 思路：按线性映射的定义验证。

\end{itemize}

\end{frame}


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\begin{frame}{习题(7.1)\#4 }

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\begin{itemize}

\item  {\color{red}问题：记 $\mathbb{R}^4$ 表示实数域上的四元列向量空间。对于 $\xi\in \mathbb{R}^4$, 定义 $\sigma(\xi)=A\xi$, 其中 $A$ 是如下矩阵，
\[ A=\begin{pmatrix} 1&-1&5&-1 \\ 1&1&-2&3 \\ 3&-1&8&1 \\ 1&3&-9&7 \end{pmatrix}. \]
求线性映射 $\sigma$ 的核空间和像空间。
}

\vspace{0.3cm}

\item 思路：核空间是齐次线性方程组的解空间。像空间是矩阵 $A$ 的列空间。


\end{itemize}

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%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{习题(7.1)\#5 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red}问题：设 $V$ 和 $W$ 是两个实向量空间。设 $\dim V=n$. 设 $\sigma:V\to W$ 是一个线性映射。
记 $\text{Ker}(\sigma)$ 和 $\text{Im}(\sigma)$ 分别是 $\sigma$ 的核空间和像空间。证明：
$$\dim \text{Ker}(\sigma) + \dim\text{Im}(\sigma) = n. $$
}

%\vspace{0.3cm}

\item 思路：取 $\text{Ker}(\sigma)$ 的一个基，将其扩充为 $V$ 的一个基。

\end{itemize}

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